If you know two sides of a right triangle, you can find the third by using the
Pythagorean Theorem: a^2 + b^2 = c^2. (back to trig page)
The four triangles have been moved around, but the "uncovered" yellow area is the same in both pictures below; a^2 + b^2 on the left and c^2 on the right.
This cool picture got me my job; I used it in my interview "mini-lecture."
- Some examples of [a, b, c] "Pythagorean Triples" are:
- [3, 4, 5] because 3^2 + 4^2 = 5^2 (check that 9 + 16 = 25.). Others are:
- [5, 12, 13] , [6, 8, 10] , [9, 12, 15] , [8, 15, 17] , [7, 24, 25] , and [20, 21, 29].
- The list goes on forever, and there are formulas to help generate Pythriples too:
- If m > n are natural nos, then if a = m^2 - n^2 , b = 2mn , c = m^2 + n^2 ,
- then [a, b, c] is a Pythag Triple (a^2 + b^2 = c^2 is guaranteed by algebra).
To solve for unknown triangle parts, you can often use the Law of Sines:
In a triangle ABC, with angles A, B, C opposite sides of lengths a, b, and c :
(sin A) / a = (sin B) / b = (sin C) / c . (back to trig page)
Proof: What can we do if there's no right angle? Make one! If we want to relate the sides a, b, c with the angles A, B, C in the diagram below, we drop a perpendicular from A to a base point D along side a ...
... now we have a right triangle ADB , so sin B = AD/AB . But AB = c , so we get the vertical dotted line AD = c sin B. Now AD is opposite both angles B and C , so we write it two ways: AD = c sin B = b sin C . By dividing we get
THE LAW OF SINES : a / (sin A) = b / (sin B) = c / (sin C) .
We really only proved the last equality but the other part works exactly the same way.
If you know two sides and the included angle, there's the Law of Cosines:
In a triangle ABC, with angles A, B, C opposite sides of lengths a, b, and c :
c^2 = a^2 + b^2 - 2ab cos C . (back to trig page)
This is cool because if C is a right angle then cos C = 0 and we get the old Pythm.
Here's an animation I did in Mathematica (to QuickTime to GIF Builder) to show two sticks of length 3 and 4; the angle changes from 0 to 180 degrees and the distance between the tips changes as shown. Think of a rubber band changing length as the sticks spread apart!
Remember you can find c by using c^2 = a^2 + b^2 - 2ab cos C.
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